This activity is intended to supplement Trigonometry, Chapter 3, Lesson 1.
Problem 1 – Proving cos2θ+sin2θ=1.
- Apply the Pythagorean Theorem to the right triangle:
- Define the right triangle trig ratios for the triangle in terms of sine and cosine:
- Substitute x=cosθ and y=sinθ into your equation from step 1:
Problem 2 - Proving sec2θ=1+tan2θ.
Select PROVE ID 2. Label the smaller triangle.
- Cross multiply and state what X and Y are equal to:
- Now substitute these values into the Pythagorean Theorem:
Problem 3 – Numerical verification
Now that we have proved the two identities using our algebra skills, it is always nice to use the power of the calculator lists to numerically verify the two identities.
- As you move the cursor around the circle, state what patterns you notice between the x−and y−values.
- What would you type into the top of a list (a formula) to numerically verify this Pythagorean Identity?
- Enter this in to the top of L4. State your results:
- What would you type into the top of a list to numerically verify this Pythagorean Identity?
- Enter these in to the top of L5 and L6. State your results:
- Explain why you cannot verify this identity numerically using the lists of the calculator.
- What alternative method can you use to verify the identity?
Problem 4 – Verifying trig identities using graphing
You can check any identity with this method.